# Questions tagged [complex-dynamics]

This tag is for questions relating to complex dynamics, study of dynamical systems defined by iteration of functions on complex number spaces. It was an area of research established by Fatou and Julia towards the beginning of the last century.

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### no wandering domain theorem for circles

I am trying to understand the proof of the no wandering domain theorem from Beardon's iterations of rational functions and thought a good start would be to omit the quasiconformal structures part and ...
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1 vote
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### Is $ds^2 = g_{11} dx^2 + 2g_{12}dxdy + g_{22}dy^2$ symmetric in the tangent vectors?

It's been a while since I studied differential geometry, so I forgot a lot of basic things. I got therefore stuck at a specific sentence in Milnor's book on complex dynamics. It says the following: A ...
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### Homeomorphism of $\mathbb{S}^1$ of certain type

Suppose $f(z)=z^2\frac{z-a}{1-\overline{a}z}$, where $a\in\mathbb{C}$. Prove that $f:\mathbb{S}^1\to\mathbb{S}^1$ is a homeomorphism if and only if $|a|\geq3$. My idea: Surjectivity is quite evident. ...
176 views

### Test to determine if a point is inside a cardioid whose cusp is not at the origin

The Mandelbrot set is the set of all complex numbers $c$ that cause the function $z_{n+1} = z_n^2 + c$ to remain bounded within a circle of radius 2 when iterated from $z_0 = 0$. Plotted in the ...
108 views

### Smooth Julia Set

My textbook tells me that for a polynomial $P$, its Julia set $J(P)$ is the nuit circle iff $P(z)=az^n$,where $\lvert a\rvert =1$,and $n\geq 2$. So I want to know whether there is a Julia set of a ...
1 vote
68 views

### Simplified expression for centers of period-three Mandlebrot bulbs

The Mandelbrot set contains three regions (two bulbs and a cardioid) with periods of three. These regions each contain a fixed point which is a root of the expression $x^3 + 2x^2 + x + 1$. The fixed ...
339 views

### Finding attractors / fixed points for the circumference of the main bulb of the Mandelbrot Set

The Mandelbrot set is the set of all complex numbers $c$ that cause the function $z_{n+1} = z_n^2 + c$ to remain bounded when iterated from $z_0 = 0$. Plotted in the complex plane it includes a main ...
142 views

### How to estimate distance from root to nearest immediate basin boundary for Newton's method in one complex variable?

Context: I want to check that the atom domain size estimate is smaller than the inradius of the Newton immediate basin, for centers of hyperbolic components in the Mandelbrot set, and thus justify ...
1 vote
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### Determine whether a point $c$ is in a wake $W_P$.

One can draw wakes in the parameter plane of the Mandelbrot set, by tracing external rays inwards from near $\infty$ (with Newton's method or other algorithm) to get polygonal outlines which can be ...
1 vote
113 views

### Attracting or parabolic cycles other than fixed points

I am studying the following complex polynomial $$P(z) = \frac{2z^{4}-2z^{3}+2z^2-z}{2z^{3}-2z^{2}+3z-2}$$ and I would like to know, if there are attracting or parabolic cycles for $P(z)$ different ...
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### Looking for an example of a function with a Julia set of positive Lebesgue measure.

I am looking for an example of a function with a Julia set of positive Lebesgue measure. I am certain these exist but cannot find a single example in the papers I have looked at. The function does not ...
1 vote
54 views

### Why is local connectivity important for polynomial Julia sets?

I'm trying to understand why local connectivity is important. I seem to remember a result that if the Julia set is locally connected then every external ray lands. I think this should mean we can get ...
1 vote
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### A Quadratic Thurston Function Has Two Distinct Critical Points

My definitions are as follows. $\hat{\mathbb{C}}$ is the Riemann sphere. Degree of a continuous map $f:\hat{\mathbb{C}}\rightarrow \hat{\mathbb{C}}$ is how many times it wraps around $\hat{\mathbb{C}}$...
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### How can I reconstruct a Julia set by a given image?

Basically, I have two images of Julia sets I liked most from a google query $\ \ \$ I want To be able to produce similar images, for that I need at least a palette from these images. To know the ...
1 vote
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### The values of the iteration $z^2 + c$ for c inside the Mandelbrot set

I'm wondering for which $c \in M$ (where M is the Mandelbrot set) the sequence $((z \to z^2 + c)^k(0))_k$ has a subsequence converging to $0$. One trivial solution is $c = 0$, though I cannot find any ...
The first image below shows the Julia Set at $-1.749512 + 0i$ (close to the base of the Period-3 Cardioid), and I'd like to find the periodic point located at where the white arrow is pointing at. ...